Well, a big problem with this requirement is that DNA isn't sequenced for identification. That's far too expensive (for now), but would allow for accurate identification excluding mosaics and twins (the former is likely underestimated in frequency since it's rarely relevant outside of this sort of analysis).
The traditional method is to chop DNA at known uncommon sequences so you get several pieces, run them on a gel that separates them according to size, and see if the sizes and number of fragments match-up. This works because humans have a few variable length repeats that vary in size and change with each generation (an over-simplification, e.g. the repeats often expand if the mother has the gene but not the father). Modern DNA analysis is a bit more sophisticated, but the underlying principle is the same.
So, how frequent are false positives? In
an analysis of Arizona's 65,000 inmates researchers found 122 9/13 matches, 20 10/13, 1 11/13, and 1 12/13. Some of these were relatives but it's hard to say how many given the study was anonymous. So, it's a low rate but not low enough to use as police would like. I'm sure it'd be very easy to find some DNA at a crime scene, run the DNA search, find one person that matches and lives in the area, and arrest them for the crime.
It's hard to argue that it's a false positive if you live a block from the crime scene and fit the physical description, but merely because people don't understand statistics very well. (E.g. if it turned out to be an 80 year old Chinese lady and not a 20-ish Black guy that resembled the description, then nobody would arrest her.) And, prosecutors are going to argue the one in 108 billion theoretic odds, without any deeper understanding of the statistics and genetics that make false positives more likely.
Setting aside, for a moment, that I have tons of non-genetics issues with such a law, I must say this is premature. In a few years we'll be able to cheaply sequence DNA and have far more accurate identifications. Furthermore, we may be able to find genes and such that make violent behavior more likely, thus aiding research. So it's illogical to adopt this technology now rather than when it actually works well in a few years. It's not like the military immediately started commissioning Wright flyers as bombers and scouts.